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I am aware that in the classical approximation of electromagnetic waves, waves can be linearly polarized (so that the B-field oscillates in one dimension as $B=B_0\cos(\omega t)$), circularly polarized (so that the field is $B=B_0(\cos(\omega t)\hat i+\sin(\omega t)\hat j)$, or elliptically polarized. My question is: why are these the only options? For instance, I know that circularly/elliptically polarized light can be generated by putting two antenna perpendicular to each other - couldn't putting three antenna perpendicular to each other generate a more complex pattern? I'm especially interested in patterns that involve the magnetic field vector moving in three dimensions, rather than just one or two.

Please let me know if there is any research that has already looked into this kind of thing - I haven't been able to find any on my own!

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  • $\begingroup$ If the wave is monochromatic there are no more exotic possibilities, this is shown in every EM book. No matter how complex the setup, you still only have two amplitudes and two phases to play with (and the overall amplitude and phase don't matter). $\endgroup$
    – Javier
    Oct 24, 2022 at 20:01

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In free space for a plane wave, the natural thing is to consider the wave as a superposition of linear and horizontal polarizations or as a superposition of right handed and left handed circular polarizations. Depending on what you are doing one is usually more convenient than the other and those kind of descriptions are complete. You can also mathematically transform between the two descriptions.

However, that doesn't mean that polarization isn't very complicated in other situations. For example when you have a guided wave in a dielectric waveguide, and depending on the geometry of the waveguide you have what are called hybrid modes and instead of having just a Transverse Electric Wave or Transverse Magnetic Wave these hybrid modes can also have an axial components of the electric and magnetic fields.

Another case where the polarization is complicated is light traveling through and anisotropic crystal where instead of the refractive index being the same in all directions, it may be uniaxial with two different indexes of refraction or biaxial with three different indexes of refraction. This is usually described by an index of refraction ellipsoid. Since the relative phase is changing as the light travels through the crystal the type of polarization is also changes as the light goes through the crystal. These properties are sometimes used to make specialized optical components such as 1/4 and 1/2 wave plates and other types of polarizers for precision optical applications.

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